A360395 Intersection of A026430 and A360394.

1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, 66, 69, 73, 78, 81, 87, 91, 96, 99, 103, 108, 114, 117, 121, 126, 129, 135, 139, 144, 150, 153, 159, 163, 168, 171, 175, 180, 186, 189, 195, 199, 204, 210, 213, 217, 222, 225, 231, 235, 240, 246, 249, 255

Offset

1,2

Comments

This is the second of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:

(1) u ^ v = intersection of u and v (in increasing order);

(2) u ^ v';

(3) u' ^ v;

(4) u' ^ v'.

Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.

For A360395, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:

u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430

u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u

v = u + 1 = A285954, except its initial 1

v' = complement of v.

Links

Table of n, a(n) for n=1..57.

Example

(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Mathematica

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Intersection[u, v]    (* A360394 *)
Intersection[u, v1]   (* A360395 *)
Intersection[u1, v]   (* A360396 *)
Intersection[u1, v1]  (* A360397 *)

Crossrefs

Cf. A026430, A356133, A360392-A360394, A360396-A360405.

Sequence in context: A020147 A316031 A316032 * A316033 A203468 A316034

Adjacent sequences: A360392 A360393 A360394 * A360396 A360397 A360398

Keyword

nonn,easy

Author

Clark Kimberling, Feb 05 2023

Status

approved